ENSENANZA Revista Mexicana de Fısica E58 (2012) 1–6 JUNIO 2012

Experimental and simulated exploration of structural deflections andacoustic waves of guitar top plates

J. Alejandro TorresCentro de Fısica Aplicada y Tecnologıa Avanzada, Universidad Nacional Autonoma de Mexico Campus Juriquilla,

Boulevard Juriquilla No. 3001 Juriquilla, Queretaro, 76230, Mexico.

J. Luis Villarreal and R. RamırezDepartamento de Visualizacion Cientıfica

Direccion General de Computo y de Tecnologıas de Informacion y ComunicacionUniversidad Nacional Autonoma de Mexico, Mexico.

Recibido el 31 de enero de 2011; aceptado el 16 de enero de 2012

The focus of this paper is to describe complex theoretical concepts about vibration and acoustics, considering experimental and computationaltools. Some modal behaviors of a structure were explained. First, an experiment was designed to naked eye visualization of mode shapesin a thin film. Later, simulated deflections of a guitar top plate model were explored. This exploration was done through original scientificvisualization software. Moreover, the developed computational tool was essential illustrating behaviors that used to be hard to visualize:acoustic waves. Sound radiation data were calculated from the vibration of the top plate model. These data were sliced to detect acousticwavefronts and directivity patterns. Experimental and computational procedures developed in this work were useful illustrating vibroacousticbehaviors. Mode shapes and natural frequencies in the thin film experiment were clearly detected. Scientific visualization software showedability to display vibroacoustic information in high density, with much semantic substance and time savings. The experiment can be easilyimplemented in a classroom, and the software can be used to explore other kind of data in standard Visualization Tool Kit format.

Keywords: Guitar; mode shapes; teaching physics.

PACS: 43.40.At; 68.60.Bs; 01.50.Pa

1. Introduction

Visualization is an indispensable tool analyzing and interpret-ing complex three-dimensional dynamics in physics [1]. Thishas been remarked in acoustics and vibrations. For example,Science magazine [2] has highlighted related animations [3]using guitar deflections as illustration. Some authors haveused the guitar as tool to teach about vibroacoustic concepts.Structural and acoustical behaviors have been already suc-cessfully explained through resonances in strings, the guitarbody and the air cavity.

For example, Boullosa [4] published historical and tech-nical notes about the acoustic guitar, scoped to undergradu-ated students. He discussed topics related with waves includ-ing shape and boundaries of a plucked string, as well as somemode shapes of a guitar top plate.

French is another researcher that has used guitars to illus-trate structural dynamics and acoustic concepts. In the firstpaper [5] of a series of four, he put the guitar string as anexample of the simplest continuous structure, explaining an-alytically about dynamic behavior. In the second paper [6] ofthe series, he describes the behavior of the guitar body using asimple acoustic-structure model, arguing that undergraduatedvibrations classes must not be focused in structural vibrationsexclusively.

1.1. The acoustic guitar

The basic behavior of the acoustic guitar radiating sound iscurrently understood. Descriptions of this process can befound, for example in Ref. 7. Briefly, the guitar works in

the following way: the player plucks the strings, put themto vibrate. A small amount of string vibrations are transmit-ted to the guitar body mainly (but not exclusively [4]) via thebridge. In turn, the guitar body oscillations, including thesoundhole air, drive the surrounding air. Sound quality of theinstrument highly depends of the interaction of the air withthe structure, mainly with the top plate.

Guitar top plates share some similar modes. But modalfrequencies do exhibit considerable variations from a modelto another. Then, the radiation efficiency of every modechanges;i.e. a mode that radiates well at determined fre-quency may lose this feature if it vibrates at another fre-quency in other guitar. In consequence, guitar sound qualitywill be different.

Nowadays, several experimental and simulated tech-niques are applied studying the vibroacoustic behavior of topplates [8]. The top plate is considered the most important partof the guitar. A slight structural modification on the top platedesign used to cause detectable changes in the vibroacous-tic behavior. Therefore, the top plate must be carefully de-signed. Analyzing the behavior of different top plate designsis a good choice to illustrate vibrations and acoustic topics.

However, naked eye visualization of the vibroacoustic be-havior in a real guitar is not possible. To see waves in solids,generally one must to detect what is happening in the geomet-rical limits of the structure, but deflections of the guitar bodyare too small. In addition, acoustic waves through the air areinvisible. Both limitations hinder an intuitive interpretationof the basic phenomenon.

2 J. ALEJANDRO TORRES, J. LUIS VILLARREAL AND R. RAMIREZ

FIGURE 1. Analogy for the lowest mode shapes in a guitar topplate, through a soap membrane vibrating in its first (top) and sec-ond (bottom) modes.

In the present paper, two tools were developed to visual-ize features related with bending and acoustic waves proper-ties. One consisted in an experiment involving a soap mem-brane to show vibrational behaviors, and the other one in soft-ware to scientific visualization of data of discrete models oftwo guitar top plates.

2. Experiment

The objective of the experiment described below is naked eyevisualization of the most prominent deflections of a vibrat-ing structure, called mode shapes [9]. It can be achieved ina classroom. The experiment only requires soap solution towash dishes; and two rods jointed by thread, forming a rect-angular frame. Two pencils for this latter task can be useful,but bigger is better. The rods together are submerged and ex-tracted from the solution. A soap membrane is created oncethe frame isquickly extended. Comparing with a guitar topplate, the membrane is very flexible and it has very low massencouraging two characteristics: mode shapes at very lowfrequencies (even less than one cycle per second) and visi-ble deflections of the surface (in the order of centimeters).Both characteristics are desirable due the objective of the ex-periment that it is being proposed.

Hand shaking slowly the membrane, the first mode shapecan be easily distinguished. All the membrane surface movesin phase and in the same direction (Fig. 1 top). If the shakingfrequency is increased, other higher mode shapes can be de-tected. For the second mode shape, a nodal line (a line with-out deflections) crossing the membrane is detected (Fig. 1bottom). A video performing this experiment is available inRef. 10. Although the effect of air in the film is, by far, largerthan for guitar top plates, explained features results be analo-gous. Comparing the explained modes of the membrane withthe lowest mode shapes of a mounted top plate (Fig. 1 right).

3. Numerical model

In order to highlight differences in behaviors of top plateswith structural modifications, a top plate with two kinds of

FIGURE 2. Steps for data calculation and visualization.

fan bracing were used in this work: a symmetrical andother one with asymmetrical design. Methodology is brieflyschematized in four steps (Fig. 2), which the last step wasbolded to remark the work developed in the present paper. Infact, the whole calculated data can be obtained using batchfiles completely included in Ref. 11.

The first step refers to simulated modal analyses per-formed using commercial finite element (FE) software. Theoutline of the top plate was restricted to zero displacements(similar to the condition mounted in the soundbox). This sim-ulation has shown good accuracy comparing with experimen-tal measurements, as it was previously reported in Ref. 12.

In the second step, it is indicated that sound radiation ofthe top plate was calculated from mode shapes obtained inthe first step. Calibrated numerical methods based on FourierAcoustics were used for this purpose. Details of the com-plete algorithm can be found in Ref. 13. Sound radiated bythe mode shape of the highest frequency was calculated toshow features of the data related with acoustic waves.

For the third step, a Matlab function was written to arraythe data to be visualized. Input files of this function mustbe individual text data in a row. Output files are arrayed inthe standard format of the Visualization Toolkit format (vtkfiles) [14]. This open-source format allows multidimensionaldata handling; through structured and unstructured points,nodal conectivity and fields, among many others specifica-tions.

In the four step, scientific visualization tools were appliedexploring the previously calculated vibroacoustic data of theguitar top plate. Procedures related with this last step are de-tailed in Subsec. 3.1.

3.1. Scientific visualization software

Scientific visualization is concerned with exploring verylarge data sets and information graphically to gain a quickunderstanding. The difference with presentation graphics is

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EXPERIMENTAL AND SIMULATED EXPLORATION OF STRUCTURAL DEFLECTIONS AND ACOUSTIC WAVES. . . 3

that latter is primarily concerned with the communication ofalready understood results [15].

Software to take advantage of the capabilities of the Sci-entific Visualization Observatory IXTLI of the UniversidadNacional Autonoma de Mexico was developed. Moreover,this software can be configured to work under Windows plat-forms equipped with graphical card.

The developed software was used loading mode shapesof both top plates and sound radiation data. Visualization ofthe vibroacoustic phenomena was done exploring six dimen-sions:

a. Three-dimensional space.Immersive virtualreality was applied using active stereoscopy (synchro-nizing projectors and stereoscopic lens). Also, this fea-ture can be turned off to use the software in a standardmonitor.

b. Vibroacoustic behavior.One can choose ifyou want explore results of structural or acousticaldata. Calculated vibrations of the top plate were shownadjusting the deflections scale and/or color mapping.Sound pressure data in the air was visualized with colormapping using slices with real-time adjustable positionand inclination.

c. Resonant frequencies.Natural frequenciesand mode shapes were extracted on each top plate de-sign, as well as sound radiaton of each one. The de-veloped software allows loading every mode shape ofboth top plates simultaneously.

d. Time.Traveling waves across the air were ob-tained by numerical methods. Pressure dataP was ex-pressed in frequency domain;i.e. specifying magni-tude and phase (which can also be written through realand imaginary part) for a fixed frequencyω [16]. Thephysical interpretation of motion from frequency do-main complex quantities is not trivial. To clarify thisbehavior, time domain visualizations were performed.The periodT of the instantaneous sound pressurept

at a fixed frequencyωq was stepped by ten instants.Traveling waves expressed in complex numbers werevisualized calculating each time step applying

pt(xl, ym, zn) = Re[P (xl, ym, zn, ωq)] cos(ωqt)

+ Im[P (xl, ym, zn, ωq)] sin(ωqt), (1)

where (xl, ym, zn) gives the spatial ubication of thepoint over the top plate, and the instantt is calculatedusingt = rT/10 for r = 1, 2, ..., 10.

4. Results and discussions

Adjusting a particular dimension in data of vibroacousticphenomenon used to cause undesirable delay in explorationsusing standard software. The developed software in this work

is capable of conveying a great amount of data of high se-mantic content in condensed form. These features representa significant improvement comparing with post-processors innumerical solutions software. Some examples are exposed inSubsec. 4.1 and 4.2.

4.1. Mode shapes exploration

Real time rotation of different three-dimensional models(both top plate and air) was possible at the same display us-ing the software developed in this work. As it can be seen inFig. 3, the software developed for the present work let us todisplay at the same screen several information of the struc-tures. Fan bracing configurations and how behaves the topplates at two different natural frequencies can be easily com-pared. At the left part of the screen is located the toolbar.User can open single windows to load it vtk models (visorbutton) or windows provided of slicing planes. In the bot-tom part of the toolbar, user can change the color mappingof the deflected top plate or choosing the mode shape to bedisplayed.

Figure 3 shows the developed software comparing bothtop plates and their respective mode shapes in an array of tworows and three columns. Each fan bracing design is shown inthe first column, the symmetrical and the asymmetrical. Inthe other columns, their respective mode shapes and natu-ral frequencies are displayed. Normal deflections to the topplate plane were overscaled and colored mapping (but thereare other options available for coloring, as grayscale).

The lowest mode shapes were handled in windows of thecentral column of Fig. 3. The vibration area is confined to thewide part of the top plate, clearly delimited by the transversalstruts. Deflection of the first mode shape for both designs isvery similar to the first mode shape of the soap membrane ofthe experiment suggested in Subsec. 2 (Fig 1 top). Moreover,the resonance frequencies were equal.

In the third column were loaded mode shapes with twonodal lines crossing the vibration area. Deflections for themode shape at 267 Hz for the symmetrical design are shown.The most similar mode shape with two nodal lines in theasymmetrical design was found at 374 Hz. However, bothmode shapes still being clearly different. Notice how barsof the fan bracing impose nodal lines. Top plate attachmentschange mass distribution and stiffness of the structure. Vari-ations in both features (mass and stiffness) caused particularresonant frequencies and mode shapes for each top plate.

These kinds of comparisons in standard numerical soft-ware are hard to achieve, because often common post-processors can not load different models to be handled at thesame screen by the user. This limitation were analyzed inRef. 17.

4.2. Sound radiation exploration

Exploring acoustic waves is more complicated than exploringbending waves. To detect wavefronts (spherical surfaces of

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4 J. ALEJANDRO TORRES, J. LUIS VILLARREAL AND R. RAMIREZ

FIGURE 3. Scientific visualization software displaying two diferrent top plate designs (left windows) and some fully turnables mode shapes(center and right windows).

of points having the same phase [18]) crossing air, internalinspection of the medium is necessary. This detection is ahard task, because visualizing the whole data is not possible,at least for a single display. An analogy of this complicationcan be done figuring a brick of Neapolitan ice cream. Fla-vors in the ice cream take the place of sound pressure values:vanilla flavor would correspond to no pressure fluctuation,chocolate to compress, and strawberry to expansion. You cansee at the outside of the brick, but you can not locate wheremore chocolate, vanilla or strawberry inside are. It would benecessary slicing the brick in a determined zone to inspectionit. Slicing is one of several applied techniques to visualize in-side of this kind of brick data.

The quick update of a model for dynamic visualizations isalso difficult and often beyond of the capabilities for currentstandard software. A time domain animation used to requirefroze the model in sequential images to obtain a movie, cre-ating a file incapable of being three-dimensionally rotated.Interaction only with frozen models is inefficient and notadequated in exploration. If it is necessary to explore an-other perspective in the model, you would have record an-other movie, generating an undesirable delay of the analysisprocess.

The time domain performance of the developed software(allowing real time interaction of dynamic data) facilitatesa better understanding of wave propagation behaviors, help-ing to study multidimensional information. In point of fact,exploration of the dynamical and kinematical performancein the acoustic data exhibit sufficient accuracy and a faithfuldisplay for relevant propagation characteristics, by means ofcombining use of software and techniques proposed in thepresent work.

Figure 4 shows a software screen with sound radiation by

a mode shape at 738 Hz. At left is located the toolbar. Atits top is located the controls of the dynamic animation ofthe data. Below these controls, the colorscale bar is plottedfor compression and expansion values of the air pressure. Atthe center of the toolbar, it is a medium-size window withthe edges of a brick sliced by three orthogonal planes. Thismedium-size window is the control for exploring the data inthe big windows (identified usinga and b). Bottom of thetoolbar have a small window which brings the whole data tothe selected big window to exploration. Sound radiation wascalculated on an air brick of 1.05 m over the top plate.

Explorations detailed in the next paragraph can be donehandling interactively a window, but in the present paper twowindows (Fig. 4(a) and Fig. 4(b) are plotted due that frozenimages will be explained).

Spherical wavefronts.In Fig. 4(a) a slice is in contactwith the structural source. The co-ordinate system indicatedin the top plate with the symmetrical design without deflec-tions shown in Fig. 3 has been preserved. Hard contrast be-tween colored pressures can be basically interpreted as phasechange (expansion to compression and vice versa). Spheri-cal wavefronts of the acoustic waves are distinguished on thepressure data on orthogonal slices to the source plane. Thelocation of the slicing planes allows seeing the whole air incontact with the top plate.

Directivity. In Fig. 4(b) the brick data has been turned.Orthogonal slices were moved to the co-rodinate system ori-gin. Sound radiation is preferentially generated forx andz postitive values, as it was marked. In the side of moresound radiation (x andz positive), wavefronts almost reachthe source plane; unlike the less sound radiation side (x neg-ative andz

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FIGURE 4. Scientific visualization software displaying the air brick sliced.

positive), where wavefronts quickly fade. This is due to theair-structure coupling. For a better insight of the phenomena,a video showing time-domain sound radiation was uploadedto the web [20]. Guitar mode shapes linked with acousticwavelengths is beyond the scope of this paper, but it was de-tailed in Ref. 13.

4.3. Performance of developed tools

The proposed experiment of Sec. 2 has been shown in lec-tures scheduled in national guitar festivals, which musiciansand luthiers represent the majority in the audience. The ex-periment represents an easy and quick way to introduce vi-bratory concepts, mainly related with modal analysis. Oc-casionally the uploaded video [10], which the experiment isshowed, receives comments by students, asking informationto perform it themselves in classroom.

Concerning to scientific visualization, the developed soft-ware has been also handled in public lectures. Structuraldata was explored by the authors using preliminar versionsof the software. For example, in a lecture [21] for acous-tic researchers, advantages comparing several mode shapesof a guitar top plate were showed. For the case of acous-tic data, some visualizations were uploaded at the end of an-other video [19]. It shows some explorations exposed duringthe doctoral dissertation exam by the first author, scheduledin the Scientific Visualization Observatory IXTLI of the Uni-versidad Nacional Autonoma de Mexico. Currently, this soft-ware can be used in IXTLI with a previous booking.

Updated version of developed software has been suc-cessfully handled using a computer Intel Pentium VI HTat 2.8 GHz and 2 GB in RAM, equipped with graphicalcard ATI Radeon HD 2600. It allowed that scientific visu-alizations were judged by luthiers in Paracho, Michoacan,

Mexico. They agree with the fact that some guitar makersusually only consider the structural vibration without keepin mind interaction with air, but importance of linking modeshapes with sound radiation was strongly reinforced in somelectures through visualizations described in this paper.

5. Conclusions

Two novel ways to exemplify vibroacoustic behaviors wereintroduced: an experimental, hand shaking a soap membrane;and the other one computational, exploring results of a gui-tar top plate model through original software. The membraneexperiment exhibits some advantages, including an easy im-plementation in almost everywhere. The experiment allowsnaked eye detection of basic behaviors of vibrations in solids:mode shapes and natural frequencies. Complexity and nov-elty of modal concepts used to cause difficult in learning bystudents. The experiment is a very interesting alternative fora classroom or physics laboratory.

Regarding the developed software, their available optionsfacilitate an intuitive interaction with a great amount of data.Several mode shapes and sound radiation data can be loadedat the same display. The information extracted from vibroa-coustic simulations can be explored using scientific visualiza-tion tools for multidimensional handling of six dimensions:three spatial dimensions (using immersive virtual reality), vi-broacoustic behavior (sound pressure or top plate deflection),resonant frequencies, and time. Interacting with all these di-mensions represents a challenge for post-processors modulesin current standard software. The developed software can beuseful exploring similar data arrayed in standard vtk format.In fact, a Matlab function was written in the present work tocreate vtk files from text data.

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6 J. ALEJANDRO TORRES, J. LUIS VILLARREAL AND R. RAMIREZ

Theoretical descriptions of structural and acousticalwaves presented in this work were reinforced exploring thebehavior of a practical case: a vibrating top plate of a guitarradiating sound. The practical and real application analyzedbrings the advantage that a lot of people can have contact withan acoustic guitar, even without strong musical instruction. Itwas shown how geometrical changes in the fan bracing de-sign modify in detectable way the mode shapes of the struc-ture. Also, spherical wavefronts and sound directivity of theplate were illustrated through a figure and two free videos.

All these experimental and numerical tools represent apowerful option to learn and to teach in sound and vibrationcourses.

Acknowledgments

The first author would like to thank to thePrograma de Be-cas Posdoctoralesof the National Autonomous University ofMexico as partial sponsor of this work.

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